BiHom-Lie brackets and the Toda equation

Abstract

We introduce a BiHom-type skew-symmetric bracket on $\mathfrak{gl}(V)$ built from two commuting inner automorphisms $\alpha =A{d}_{\psi }$ and $\beta =A{d}_{\varphi }$ with $\psi ,\varphi \in \mathfrak{gl}(V)$ and integers $i,j$. We prove that $(\mathfrak{gl}(V),[\cdot ,\cdot {]}_{(\psi ,\varphi )}^{(i,j)},\alpha ,\beta )$ is a BiHom--Lie algebra, and we study the Lax equation obtained by replacing the commutator in the finite nonperiodic Toda lattice by this bracket. For the symmetric choice $\varphi =\psi$ with $(i,j)=(0,0)$, the deformed flow is equivariant under conjugation and becomes gauge-equivalent, via $\tilde{L}={\psi }^{-1}L\psi$, to a Toda-type Lax equation with a conjugated triangular projection. In particular, scalar deformations amount to a constant rescaling of time. On embedded $2\times 2$ blocks, we derive explicit trigonometric and hyperbolic formulas that make symmetry constraints (e.g. tracelessness) transparent. In the asymmetric hyperbolic case, we exhibit a trace obstruction showing that the right-hand side is generically not a commutator, which amounts to symmetry breaking of the isospectral property. We further extend the construction to the weakly coupled Toda lattice with an indefinite metric and provide explicit $2\times 2$ solutions via an inverse-scattering calculation, clarifying and correcting certain formulas in the literature. The classical Toda dynamics are recovered at special parameter values.

22 Pages

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